| MSC: | 34A08 | Fractional differential equations |
| 26A33 | Fractional derivatives and integrals | |
| 39B99 | Difference and functional equations | |
| 45G10 | Other nonlinear integral equations | |
| 34B37 | Boundary value problems with impulses | |
| 34B15 | Nonlinear boundary value problems | |
| 34B16 | Singular nonlinear boundary value problems |
Abstract:
In this paper, we firstly prove existence and uniqueness of solutions of
Cauchy problems for nonlinear fractional differential equations involving the Caputo fractional
derivative, the Riemann-Liouville derivative, the Caputo type Hadamard derivative
and the Riemann-Liouville type Hadamard fractional derivatives of order α ∈ [n − 1, n)
respectively by using Picard iterative technique under some suitable assumptions. Meanwhile,
we get the iterative approximation solutions of these kind of Cauchy problems. Secondly we
obtain exact expression of piecewise continuous solutions of the linear fractional differential
equations. These results provide new methods to convert an impulsive fractional differential
equation to a fractional integral equation. Thirdly, four classes of boundary value problems
for singular fractional differential equations with impulse effects are proposed. Sufficient
conditions are given for the existence of solutions of these problems. We allow the nonlinearity
p(t)f (t, x) in fractional differential equations to be singular at t = 0, 1. Finally, by
establishing existence results on solvability of two class of impulsive boundary value problems
of fractional differential equations, we make a comparison on impulsive boundary value
problems for two kinds of fractional differential equations, one has a single starting point and
the other one has multiple starting points. In order to avoid misleading the readers, a
mistake in [Impulsive integral boundary value problems of the higher-order fractional differential
equation with eigenvalue arguments, Adv. Differ. Equa. (2015) 2015: 382] is also corrected.